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Question concerning Aldrovandi/Freitas-article
#5
andydude Wrote:So what is the difference between "normality" and "distinct eigenvalues"? I thought that distinct eigenvalues were sufficient for diagonizability...

Andrew Robbins

"normal": let M be a matrix (we're discussing real matrices for example). Then M is "normal", if M commutes with its transpose M*M' = M'*M
This equality is obviously true for symmetric M, but also for some others.

It is said, that for normal matrices, if
T*M*T^-1 = D , D diagonal,
then T is orthogonal, meaning T*T' = T*T^-1=I (I think T is always a rotation)
and also

T*M*T' = D

(from other context I'm used to denote rotation-matrices by letter T)
---------------------------

not "normal", but still diagonalizable (the more general case):

W*M*W^-1 = D

no specific properties on W.

Related to current discussion: if M is triangular (and diagonalizable), I think W is also triangular (but I must check this), and the eigenvalues are the entries of its diagonal.

-------------------

The question whether eigenvalues are distinct or not is not relevant here; this is only relevant for the description of further properties of W (whether it is unique ... )

Gottfried
Gottfried Helms, Kassel
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Messages In This Thread
RE: Question concerning Aldrovandi/Freitas-article - by Gottfried - 01/19/2008, 09:10 AM

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